Tessellation

Tessellation of a flat surface refers to the repeated placement of shapes with no overlaps and no gaps. These shapes are also called tiles. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.

In math

Six regular polygons and a few other shapes, making tessellations.

Only three regular polygons can cover a perfectly level surface: triangles, squares, and hexagons. Other shapes, like pentagons, will need help from other shapes like rhombi.

Penrose tilings are special cases where the pattern never repeats, no matter how much it continues.

In real life

Since ancient times, people have used tessellations to decorate buildings.

The artist M.C. Escher was brilliant at painting pictures involving tessellation. He was inspired by the walls and floors in Muslim buildings.^[1]

Tessellations appear a lot in nature. One of the best examples is the honeycomb. Honeycombs are built with hexagonal holes to hold honey and bees. If you can build a square that's big enough to hold 225 bees, you can bend the edges into a hexagon and hold 260 bees. If you want to have a lot of area with only a certain amount of perimeter, the best shape is a circle, so bees build circles, and the wax slowly melts into hexagons.^[2] This is called the honeycomb theorem.^[3]

All of the buildings in this picture of Belfast in Ireland are rectangular to fit together, with some spaces for alleyways.
This bee can hold more honey with the hexagon walls of its honeycomb than with square walls.
The James Webb telescope has 18 big hexagonal mirrors.
The yolks of these eggs fit together like hexagons due to the way that circles tesselate.
The same thing seems to happen with these bubbles.

In puzzles

Many puzzles like Tangram are built around tessellation.

This square is made of seven pieces...
but with a little imagination, it can be changed into a number of shapes like this one.
This game is called Tetris, and the goal is to fit plenty of shapes together.
This rectangle is made of 12 pentominoes, which are in turn made of 9 smaller pentominoes, so all together, the rectangle has 108 small pentominoes.

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↑ Locher, J. L. (1971). The World of M. C. Escher. Abrams. pp. 17, 70–71. ISBN 0-451-79961-5.
↑ Wentworth, Thompson, D'Arcy (1917). On growth and form. Cambridge [Eng.]: University press.{{cite book}}: CS1 maint: multiple names: authors list (link)
↑ Hales, T. C. (2001-01-01). "The Honeycomb Conjecture". Discrete & Computational Geometry. 25 (1): 1–22. doi:10.1007/s004540010071. ISSN 1432-0444.

[1] Locher, J. L. (1971). The World of M. C. Escher. Abrams. pp. 17, 70–71. ISBN 0-451-79961-5.

[2] Wentworth, Thompson, D'Arcy (1917). On growth and form. Cambridge [Eng.]: University press.{{cite book}}: CS1 maint: multiple names: authors list (link)

[3] Hales, T. C. (2001-01-01). "The Honeycomb Conjecture". Discrete & Computational Geometry. 25 (1): 1–22. doi:10.1007/s004540010071. ISSN 1432-0444.

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